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($ABRIDGED$) We probe the physical properties and large-scale environment ofradio AGN in the faintest FR population to-date, and link them to their radiostructure. We use the VLA-COSMOS Large Project at 3 GHz, with resolution andsensitivity of 0".75 and 2.3 $\mu$Jy/beam, respectively, to explore the FRdichotomy down to $\mu$Jy levels. We classify objects as FRIs, FRIIs or hybridFRI/FRII based on the surface-brightness distribution along their radiostructure. Our control sample is the jet-less/compact radio AGN (COM AGN) whichshow excess radio emission at 3 GHz VLA-COSMOS exceeding what is coming fromstar-formation alone; this sample excludes FRs. Largest angular projected sizesof FR objects are measured by a machine-learning algorithm and also by hand,following a parametric approach to the FR classification. Eddington ratios arecalculated using scaling relations from the X-rays, while we include the jetpower by using radio luminosity as a probe. We investigate their hostproperties (star-formation ratio, stellar mass, morphology), and we exploretheir incidence within X-ray galaxy groups in COSMOS, as well as in the densityfields and cosmic-web probes in COSMOS. Our sample is composed of 59 FRIIs, 32FRI/FRIIs, 39 FRIs, and 1818 COM AGN at 0.03 $\le z \le$ 6. FR objects have onaverage similar radio luminosities ($L_{\rm 3~GHz}\rm \sim10^{23}~W~Hz^{-1}~sr^{-1}$), spanning a range of $\rm10^{21-26}~W~Hz^{-1}~sr^{-1}$, and lie at a median redshift of $z ~\sim ~1$.FRs reside in their majority in massive quenched hosts ($M_{*}~> 10^{10.5}M_{\odot}$), with older episodes of star-formation linked to lower X-ray galaxygroup temperatures, suggesting radio-mode AGN quenching. Irrespective of theirradio structure, FRs and COM AGN are found in all types and densityenvironments (group or cluster, filaments, field).

FR-type radio sources at 3 GHz VLA-COSMOS: Relation to physical properties and large-scale environment